EE3TP4_26_DTFT_System_Analysis_v3_Lecture 33

EE3TP4_26_DTFT_System_Analysis_v3_Lecture 33 -...

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X = n =−∞ x [ n ] e j n x [ n ]= 1 π π X  e jn d Discrete-Time Fourier Transform (DTFT) X [ k ]= n = 0 N 1 x [ n ] e j2π kn / N k = 0, 1, 2, . .. , N 1 x [ n ]= 1 N k = 0 N 1 X [ k ] e j2π kn / N n = 0, 1, 2, . .. , N 1 Discrete Fourier Transform (DFT) These overheads were originally developed by Mark Fowler at Binghamton University, State University of New York.
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5.5: System analysis via DTFT h [ n ] x [ n ] y [ n ]= h [ n ]∗ x [ n ] = i =−∞ h [ i ] x [ n i ] Recall that in Ch. 5 we saw how to use frequency domain methods to analyze the input-output relationship for the C-T case. We now do a similar thing for D-T Define the “Frequency Response” of the D-T system We now return to Ch. 5 for its DT coverage! Back in Ch. 2, we saw that a D-T system in “zero state” has an output-input relation of: H = n =−∞ h [ n ] e j n DTFT of h [ n ] Perfectly parallel to the same idea for CT systems!
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From Table of DTFT properties: x [ n ]∗ h [ n ] ↔ X  H  So we have: h [ n ] H  x [ n ] X  y [ n ]= h [ n ]∗ x [ n ] Y = X  H  Y ∣=∣ X ∣∣ H ∣ Y  = X  + H  So … So … in general we see that the system frequency response re-shapes the input DTFT’s magnitude and phase. System can : -emphasize some frequencies -de-emphasize other frequencies Perfectly parallel to the same ideas for CT systems! The above shows how to use DTFT to do general DT system analyses … virtually all of your insight from the CT case carries over!
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Response to Sinusoidal Input x [ n ]= A cos  0 n θ n =− 3, 2, 1, 0, 1, 2, 3,. .. From DTFT Table: X = { [ e δ  0  e δ − 0 ] π  π periodic elsewhere X  0 − 0 π π We only need to focus our attention here Y = H
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EE3TP4_26_DTFT_System_Analysis_v3_Lecture 33 -...

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